Proportional & inversely proportional

The tap is running into the bucket: after 2 minutes there are 24 litres in it, after 4 minutes 48 litres. The same evening a group splits a taxi fare of 60 euros: with two people each pays 30 euros, with four only 15 euros. Both times two quantities are linked, but they move in opposite directions. With the water, one grows along with the other. With the taxi, it shrinks. These two patterns have names: proportional and inversely proportional.

Proportional: twice as much, twice as much

Back to the bucket. After 1 minute there are 12 litres in it, after 2 minutes 24 litres, after 3 minutes 36 litres, after 5 minutes 60 litres. Double the time and the amount doubles, triple it and the amount triples. The real core, though, sits in the division: if you divide the litres by the minutes for every value pair, you always get the same number, namely 12. That number is called the constant of proportionality $k$, here 12 litres per minute. It is the thing that stays put while both quantities change.

Plot the value pairs in a coordinate system, minutes to the right and litres upwards, and all the points lie on a straight line. That line runs through the origin, because after 0 minutes there are 0 litres in the bucket. And that is exactly the test: a straight line alone is not enough, it also has to pass through the origin. The factor $k$ is the slope, so a bigger $k$ makes the line steeper. A tap delivering 20 litres per minute gives the same kind of picture, just steeper.

Inversely proportional: more of this, less of that

Now the taxi. The ride costs 60 euros no matter how many people come along, and that amount gets divided by the number of people. With two people each pays 30 euros, with three 20 euros, with four 15 euros, with five 12 euros. Twice as many people means half the amount each. The quotient is no help here, but the product is: 2 times 30, 3 times 20, 4 times 15 and 5 times 12 all give 60. The same pattern sits inside a 180 km drive: at 60 km/h it takes 3 hours, at 90 km/h only 2 hours, and speed times time stays 180.

So how do you tell which case you are in? Take two value pairs and run the doubling test: if one value doubles and the other doubles too, it is proportional. If the other one halves, it is inversely proportional. If neither happens, it is neither of the two, and that third case shows up more often than people expect. A phone plan with a 5 euro base fee plus 2 euros per gigabyte does grow steadily, but at 0 gigabytes you are already paying 5 euros. The line misses the origin, so the plan is not proportional. In a coordinate system an inversely proportional relationship looks completely different, by the way: a falling curve that only creeps towards the axes and never touches them.